Glossary of algebraic groups

There are a number of mathematical notions to study and classify algebraic groups.

In the sequel, G denotes an algebraic group over a field k.

notion	explanation	example	remarks
linear algebraic group	A Zariski closed subgroup of ${{\rm {GL}}}_{n}$ for some n	${{\rm {SL}}}_{n}$	Every affine algebraic group is isomorphic to a linear algebraic group, and vice versa
affine algebraic group	An algebraic group which is an affine variety	${{\rm {GL}}}_{n}$ , non-example: elliptic curve	The notion of affine algebraic group stresses the independence from any embedding in ${{\rm {GL}}}_{n}$
commutative	The underlying (abstract) group is abelian.	${{\mathbb G}}_{a}$ (the additive group), ${{\mathbb G}}_{m}$ (the multiplicative group),^[1] any complete algebraic group (see abelian variety)
diagonalizable group	A closed subgroup of $({\mathbb {G}}_{m})^{n}$ , the group of diagonal matrices (of size n-by-n)
simple algebraic group	A connected group which has no non-trivial connected normal subgroups	${{\rm {SL}}}_{n}$
semisimple group	An affine algebraic group with trivial radical	${{\rm {SL}}}_{n}$ , ${{\rm {SO}}}_{n}$	In characteristic zero, the Lie algebra of a semisimple group is a semisimple Lie algebra
reductive group	An affine algebraic group with trivial unipotent radical	Any finite group, ${{\rm {GL}}}_{n}$	Any semisimple group is reductive
unipotent group	An affine algebraic group such that all elements are unipotent	The group of upper-triangular n-by-n matrices with all diagonal entries equal to 1	Any unipotent group is nilpotent
torus	A group that becomes isomorphic to $({\mathbb {G}}_{m})^{n}$ when passing to the algebraic closure of k.	${{\rm {SO}}}_{2}$	G is said to be split by some bigger field k' , if G becomes isomorphic to G_mⁿ as an algebraic group over k'.
character group X_∗(G)	The group of characters, i.e., group homomorphisms $G\rightarrow {{\mathbb G}}_{m}$	$X^{*}({\mathbb {G}}_{m})\cong {\mathbb {Z}}$
Lie algebra Lie(G)	The tangent space of G at the unit element.	${{\rm {Lie}}}({{\rm {GL}}}_{n})$ ) is the space of all n-by-n matrices	Equivalently, the space of all left-invariant derivations.

Notes

↑ These two are the only connected one-dimensional linear groups, Springer 1998, Theorem 3.4.9

References

Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012
Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713

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